I do not recall ever hearing a fluent English speaker say "broke his feelings".
It is a common idiom to say that someone "broke his/her heart". Typically this is said if a wife or girlfriend or husband or boyfriend terminates a relationship. It is also commonly said if someone close to a person disappoints him in some extreme way, like "When he learned that his son had become a thief and a killer, it broke his heart."
It is also common to say that someone "hurt his feelings". This is used for a disappointment or offense on a much smaller scale than "broke his heart". Like, "When she didn't show up for their first date, it hurt his feelings." "When he learned that his son had failed a history test, it hurt his feelings."
I don't know any logical reason why we say "broke his heart" but not "hurt his heart", and "hurt his feelings" but not "broke his feelings". But that's the nature of idioms. Some phrases just come to be commonly accepted, and others ... don't.
Regarding 25 million hits for "break * feelings": I checked out the first two pages and didn't see any that were "break his feelings" or "break her feelings". Rather, there were many other words in between, like "When she broke up with him, it hurt his feelings". Finding such a high number for that pair of words doesn't surprise me at all: People routinely talk their feelings when someone breaks up with them.